Compute the proportion that were correctly or incorrectly classified.

\[\mathrm{Accuracy} = n^{-1} \sum_{i=1}^n 1({\hat{y}}_i = y_i)\]

\[\mathrm{Error Rate} = n^{-1} \sum_{i=1}^n 1(\hat{y}_i \ne y_i)\]

• Suppose in the true population, 95% are negative.
• Classifier that always outputs negative is 95% accurate. This is the baseline accuracy.
• Accuracy is not a useful measure.

Literature on learning from unbalanced classes:

http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=5299216 http://link.springer.com/article/10.1007%2Fs10115-013-0670-6 http://www.computer.org/csdl/proceedings/icdm/2012/4905/00/4905a695-abs.html http://www.computer.org/csdl/proceedings/icdm/2011/4408/00/4408a754-abs.html

npos <- 500; nneg <- 500; set.seed(1)
df <- rbind(data.frame(x=rnorm(npos,mupos), y=1),data.frame(x=rnorm(nneg,muneg),y=-1)); df$y <- as.factor(df$y)
sep <- tune(svm,y~x,data=df,ranges=list(gamma = 2^(-1:1), cost = 2^(2:4)))
df$ypred <- predict(sep$best.model)
ggplot(df,aes(x=x,fill=y)) + geom_histogram(alpha=0.2,position="identity",bins=51) + geom_point(aes(y=ypred,colour=ypred)) + scale_color_discrete(drop=FALSE)

npos <- 50; nneg <- 950; set.seed(1)
df <- rbind(data.frame(x=rnorm(npos,mupos), y=1),data.frame(x=rnorm(nneg,muneg),y=-1)); df$y <- as.factor(df$y)
rsep <- tune(svm,y~x,data=df,ranges=list(gamma = 2^(-1:1), cost = 2^(2:4)))
df$ypred <- predict(rsep$best.model);
ggplot(df,aes(x=x,fill=y)) + geom_histogram(alpha=0.2,position="identity",bins=51) + geom_point(aes(y=ypred,colour=ypred)) + scale_color_discrete(drop=FALSE)

newneg <- df %>% filter(y == 1) %>% sample_n(900,replace=T); dfupsamp <- rbind(df,newneg)
upsep <- tune(svm,y~x,data=dfupsamp,ranges=list(gamma = 2^(-1:1), cost = 2^(2:4)))
dfupsamp$ypred <- predict(upsep$best.model);
ggplot(dfupsamp,aes(x=x,fill=y)) + geom_histogram(alpha=0.2,position="identity",bins=51) + geom_point(aes(y=ypred,colour=ypred)) + scale_color_discrete(drop=FALSE)

df$upsampred <- predict(upsep$best.model,df)
mean(df$y == df$ypred)

## [1] 0.95

mean(df$y == df$upsampred)

## [1] 0.694

No upsampling: 95% Accuracy

Upsampled: 68% Accuracy

So why do you like the upsampled classifier better?

Careful! A “false positive” is actually a negative and a “false negative” is actually a positive.

• Unbalanced classes

library(caret)
cm_orig <- confusionMatrix(df$ypred, df$y, positive = "1", mode = "prec_recall")
print(cm_orig$table)

##           Reference
## Prediction  -1   1
##         -1 950  50
##         1    0   0

• Upsampled Data

cm_up <- confusionMatrix(df$upsampred, df$y, positive = "1", mode = "prec_recall")
print(cm_up$table)

##           Reference
## Prediction  -1   1
##         -1 654  10
##         1  296  40

• In Information Retrieval, typically very few positives, many negatives. (E.g. billion webpages, dozen relevant to search query.) Focus is on correctly identifying positives.
• Recall: What proportion of the positives in the population do I correctly identify?
• Precision: What proportion of the instances I labeled positive are actually positive?

\[\mbox{F-measure} = 2 \frac{\mathrm{Precision}\cdot\mathrm{Recall}}{\mathrm{Precision}+\mathrm{Recall}}\] https://en.wikipedia.org/wiki/F1_score

\[\mbox{F-measure} = 2 \frac{\mathrm{Precision}\cdot\mathrm{Recall}}{\mathrm{Precision}+\mathrm{Recall}}\] https://en.wikipedia.org/wiki/F1_score

For the “always predict -1” classifier, recall = 0, precision = 0, so
F-measure = 0.

For the classifier learned from up-sampled data,

prec <- sum(df$y == 1 & df$upsampred == 1) / sum(df$upsampred == 1)
recall <- sum(df$y == 1 & df$upsampred == 1) / sum(df$y == 1)
F1.upsamp <- 2 * prec*recall / (prec + recall)
print(F1.upsamp)

## [1] 0.2072539

NOTE that F-measure is not “symmetric”; it depends on the definition of the positive class. Typically used when positive class is rare but important e.g. information retrieval.

• Sensitivity: What proportion of the positives in the population do I correctly label?
• Specificity: What proportion of the negatives in the population do I correctly label?

\[\mbox{BalancedAccuracy} = \frac{1}{2} (\mathrm{Sensitivity}+\mathrm{Specificity})\]

https://en.wikipedia.org/wiki/Accuracy_and_precision#In_binary_classification
Note: Sensitivity is same as Recall.

• Sensitivity: What proportion of the positives in the population do I correctly label?
• Specificity: What proportion of the negatives in the population do I correctly label?

\[\mbox{BalancedAccuracy} = \frac{1}{2} (\mathrm{Sensitivity}+\mathrm{Specificity})\]

For “always predict -1” classifier, sensitivity = 0, specificity = 1, balanced accuracy = 0.5.

For the classifier learned from up-sampled data,

sens <- sum(df$y == 1 & df$upsampred == 1) / sum(df$y == 1)
spec <- sum(df$y == -1 & df$upsampred == -1) / sum(df$y == -1)
bal.acc.upsamp <- 0.5*(sens + spec)
print(bal.acc.upsamp)

## [1] 0.7442105

Recall:

\[\mbox{BalancedAccuracy} = \frac{1}{2} (\mathrm{Sensitivity}+\mathrm{Specificity})\]

What if e.g. false positives are more costly than false negatives?

Let \(\mathrm{P}\) and \(\mathrm{N}\) the proportions of positives and negatives in the population.

\[\mathrm{FNRate} = (1 - \mathrm{Sensitivity}), \mathrm{FPRate} = (1 - \mathrm{Specificity})\]

\[\mbox{NormExpectedCost} = c_{\mathrm{FP}}\cdot\mathrm{FPRate}\cdot\mathrm{P}+ c_{\mathrm{FN}}\cdot\mathrm{FNRate}\cdot\mathrm{N}\]

http://www.csi.uottawa.ca/~cdrummon/pubs/pakdd08.pdf

• Suppose classifier can rank inputs according to “how positive” they appear to be.
• E.g., can use probability from Logistic Regression, or \(w^{\mathsf T}x + b\) for SVM.
• By adjusting the “threshold” value for deciding an instance is positive, we can obtain different false positive rates. Low threshold gives higher false positives (but higher true negatives), high threshold gives lower false positives (but higher false negatives.)
• ROC curve: Try all possible cutoffs, plot FPR on \(x\)-axis, TPR on \(y\)-axis.

https://en.wikipedia.org/wiki/Receiver_operating_characteristic

Think: “If I fix FPR at 0.4, what is my TPR?”

Obviously, higher is better. Random guessing gives an ROC curve along \(y = x\).

If the area under the curve (AUC) is 1, we have a perfect classifier. AUC of 0.5 is worst possible.

Very common measure of classifier performance, especially when classes are imbalanced.

library(ROCR)
preds <- attr(predict(rsep$best.model,df,decision.values = TRUE),"decision.values")
plot(performance(prediction(preds,df$y),"tpr","fpr"))
abline(a = 0, b = 1)

performance(prediction(preds,df$y),"auc")@y.values

## [[1]]
## [1] 0.4750316

library(ROCR)
preds <- attr(predict(upsep$best.model,df,decision.values = TRUE),"decision.values")
plot(performance(prediction(preds,df$y),"tpr","fpr"))
abline(a = 0, b = 1)

performance(prediction(preds,df$y),"auc")@y.values

## [[1]]
## [1] 0.7996842

• The Area Under the Reciever Operating Characteristic Curve is also called the \(c\)-statistic (“concordance”)
• Also the statistic for the Wilcoxon-Mann-Whitney hypothesis test of equal distributions

If we care about all these measures, why do we optimize misclassification rate, or margin, or likelihood?

• Computational tractability

• Classifier learned the way we described often perform well measures presented here

• However
  • There are methods for learning e.g. SVMs by optimizing ROC
  • Cost-sensitive learning is also widespread
  • Methods are evolving; a quick google scholar search is a good idea.